A Koksma-Hlawka inequality for general discrepancy systems

Florian Pausinger*, Anne Marie Svane

*Corresponding author for this work

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.

Original languageEnglish
Pages (from-to)773-797
Number of pages25
JournalJournal of Complexity
Volume31
Issue number6
Early online date16 Jun 2015
DOIs
Publication statusPublished - 01 Dec 2015

Keywords

  • Hardy-Krause variation
  • Harman variation
  • Integral geometry
  • Koksma-Hlawka theorem

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Statistics and Probability
  • Numerical Analysis
  • Control and Optimization
  • Applied Mathematics

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